Chapter 11.12, Problem 22P

In the pendulum problem, $\theta =\alpha \sin \sqrt{g/l}t$ is an approximate solution when the amplitude $\alpha$ is small enough for the motion to be considered simple harmonic. Show that the corresponding exact solution when $\alpha$ is not small is where $k=\sin (\alpha /2)$ is the modulus of the elliptic function. Show that this reduces to the simple harmonic motion solution for small amplitude $\alpha .$